Physlib.StringTheory.FTheory.SU5.Fluxes.NoExotics.ChiralIndices

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If the condition of no chiral exotics holds for $F$, then every chiral index $\chi$ in the multiset of chiral indices for the representation $D = (\bar{3}, 1)_{1/3}$ is non-negative, i.e., $\chi \geq 0$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. If the condition of no chiral exotics is satisfied (implying the total number of anti-chiral $L$ and $D$ states is zero), then every chiral index $ci$ in the multiset of chiral indices for the representation $L = (1,2)_{-1/2}$ (defined as $M + N$ for each curve) is non-negative, i.e., $ci \geq 0$.

For a multiset $F$ of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves in an $SU(5)$ F-theory model, if $F$ satisfies the condition of having no chiral exotics (meaning the total number of anti-chiral representations for $Q$, $U$, and $E$ is zero), then every chiral index $ci$ in the multiset of chiral indices for the representation $Q = (3, 2)_{1/6}$ is non-negative, i.e., $ci \ge 0$.

For a multiset $F$ of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $10$-dimensional representation matter curves in an $SU(5)$ F-theory model, if there are no chiral exotics in the spectrum, then every chiral index $\chi(U) \in F.\text{chiralIndicesOfU}$ for the Standard Model representation $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ is non-negative, i.e., $0 \le \chi(U)$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves. For each curve, the chiral index for the Standard Model representation $E = (1, 1)_1$ is given by $\chi_E = M + N$. If the flux multiset $F$ satisfies the condition of having no chiral exotics, then every chiral index $\chi_E$ in the multiset of chiral indices for $E$ is non-negative, i.e., $\chi_E \geq 0$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ associated with the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If the condition of no chiral exotics is satisfied for $F$, then the sum of the chiral indices for the $D = (\bar{3}, 1)_{1/3}$ representation—where each index corresponds to the chirality flux $M$ of a matter curve—is equal to $3$.

For a collection $F$ of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model, if there are no chiral exotics, then the sum of the chiral indices for the representation $L = (1, 2)_{-1/2}$ is equal to 3. Here, the chiral index for each curve is given by $M + N$, and the "no exotics" condition implies that the total number of chiral $L$ and $D$ states is 3 and the total number of anti-chiral states is 0.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional matter curves in an $SU(5)$ F-theory model. If there are no chiral exotics in the spectrum, then the sum of the chiral indices for the representation $Q = (\mathbf{3}, \mathbf{2})_{1/6}$ (where each index is given by the chirality flux $M$) across all matter curves in $F$ is equal to $3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves, where $M$ is the chirality flux and $N$ is the hypercharge flux. If the condition of no chiral exotics is satisfied (meaning the total number of chiral states is 3 and anti-chiral states is 0 for the Standard Model representations $Q, U,$ and $E$), then the sum of the chiral indices $\chi(U) = M - N$ for the representation $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ over all matter curves in $F$ is equal to 3.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves. If the condition of no chiral exotics is satisfied (meaning the total number of chiral $E = (1, 1)_1$ representations is 3 and the number of anti-chiral $E$ representations is 0), then the sum of the chiral indices $\chi_E = M + N$ for the representation $E$ across all curves in $F$ is equal to $3$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If the condition of no chiral exotics is satisfied for $F$, then every chiral index $\chi$ in the multiset of chiral indices for the representation $D = (\bar{3}, 1)_{1/3}$ satisfies $\chi \leq 3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves. If the condition of no chiral exotics is satisfied, then every chiral index $ci$ in the multiset of chiral indices for the representation $L = (1, 2)_{-1/2}$ satisfies $ci \leq 3$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional matter curves in an $SU(5)$ F-theory model. If the model satisfies the condition of having no chiral exotics, then any chiral index $ci$ in the multiset of chiral indices for the representation $Q = (3, 2)_{1/6}$ satisfies $ci \le 3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $10$-dimensional representation matter curves. If the condition of no chiral exotics is satisfied (meaning the total number of chiral states is 3 and the total number of anti-chiral states is 0 for the Standard Model representations $Q, U,$ and $E$), then every individual chiral index $\chi(U) = M - N$ in the multiset of chiral indices for the representation $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ satisfies $\chi(U) \leq 3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional representation matter curves. The chiral index for the Standard Model representation $E = (1, 1)_1$ on a curve is given by $\chi_E = M + N$. If the configuration $F$ satisfies the condition of having no chiral exotics (which requires the sum of all such indices to be 3 and each index to be non-negative), then every individual chiral index $\chi_E$ in the multiset must satisfy $\chi_E \le 3$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ matter curves. If the condition of no chiral exotics is satisfied for $F$ (meaning the total number of chiral states is 3 and the total number of anti-chiral states is 0 for the Standard Model representations $L$ and $D$), then every individual chiral index $\chi_D$ in the multiset of chiral indices for the representation $D = (\bar{3}, 1)_{1/3}$ satisfies $\chi_D \in \{0, 1, 2, 3\}$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves. If the condition of no chiral exotics is satisfied (which implies the total number of chiral $L$ states is 3 and no anti-chiral states exist), then every chiral index $ci$ in the multiset of chiral indices for the Standard Model representation $L = (1, 2)_{-1/2}$ must be one of the values $0, 1, 2,$ or $3$. That is, $ci \in \{0, 1, 2, 3\}$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional matter curves in an $SU(5)$ F-theory model. If the model satisfies the condition of having no chiral exotics, then any chiral index $ci$ in the multiset of chiral indices for the representation $Q = (3, 2)_{1/6}$ must be an element of the set $\{0, 1, 2, 3\}$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with $10$-dimensional representation matter curves. If the condition of no chiral exotics is satisfied, then every chiral index $\chi(U) = M - N$ in the multiset of chiral indices for the Standard Model representation $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ belongs to the set $\{0, 1, 2, 3\}$.

In an $SU(5)$ F-theory model, let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{10}$-dimensional representation matter curves. For each curve, the chiral index for the Standard Model representation $E = (1, 1)_1$ is defined as $\chi_E = M + N$. If the flux configuration $F$ satisfies the condition of having no chiral exotics, then every individual chiral index $\chi_E$ in the multiset of chiral indices for $E$ must be an element of the set $\{0, 1, 2, 3\}$.

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z} \times \mathbb{Z}$ associated with the $\mathbf{\bar{5}}$ matter curves in an $SU(5)$ F-theory model. If the condition of no chiral exotics is satisfied for $F$, then for any sub-multiset $S \subseteq F$, the sum of the chirality fluxes $M$ (which correspond to the chiral indices of the $D = (\bar{3}, 1)_{1/3}$ representation) satisfies: \[ \sum_{(M, N) \in S} M \le 3. \]

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the $\mathbf{\bar{5}}$ representation matter curves in an $SU(5)$ F-theory model. Suppose that the condition of no chiral exotics is satisfied. For any sub-multiset $S \subseteq F$, the sum of the chiral indices for the representation $L = (1, 2)_{-1/2}$ (where the index for each curve is $M + N$) is less than or equal to 3, i.e., $$\sum_{(M, N) \in S} (M + N) \leq 3.$$

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional matter curves in an $SU(5)$ F-theory model. If the model satisfies the condition of having no chiral exotics (implying the total sum of chiral indices for the representation $Q = (\mathbf{3}, \mathbf{2})_{1/6}$ is 3 and all indices are non-negative), then for any sub-multiset $S \subseteq F$, the sum of the chiral indices of $Q$ in $S$ is less than or equal to 3: $$\sum_{(M, N) \in S} M \leq 3.$$

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with the 10-dimensional matter curves in an $SU(5)$ F-theory model. Suppose that the condition of no chiral exotics is satisfied. For any sub-multiset $S \subseteq F$, the sum of the chiral indices for the representation $U = (\bar{\mathbf{3}}, \mathbf{1})_{-2/3}$ (where the index for each curve is $M - N$) is less than or equal to 3: $$\sum_{(M, N) \in S} (M - N) \leq 3.$$

Let $F$ be a multiset of flux pairs $(M, N) \in \mathbb{Z}^2$ associated with 10-dimensional matter curves in an $SU(5)$ F-theory model. If the model satisfies the condition of having no chiral exotics (which implies that for the representation $E = (1, 1)_1$, the total sum of chiral indices $\chi_E = M + N$ is 3 and each $\chi_E \ge 0$), then for any sub-multiset $S \subseteq F$, the sum of the chiral indices for the representation $E$ in $S$ is less than or equal to 3: $$\sum_{(M, N) \in S} (M + N) \leq 3.$$